A Bloch function is a function analytic in the unit disk such that is bounded. First we generalize the theorem of Rohde that, for every "bad" Bloch function, follows any prescribed curve at a bounded distance for in a set of Hausdorff dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that does not vary much on each circle except for small exceptional arcs. We show e.g. that
holds either for all or for none.
Cite this article
Juan J. Donaire, Christian Pommerenke, On radial behaviour and balanced Bloch functions. Rev. Mat. Iberoam. 15 (1999), no. 3, pp. 429–449DOI 10.4171/RMI/261